A robust proof of the instability of naked singularities of a scalar field in spherical symmetry

Published in 1999, Christodoulou proved that the naked singularities of a self-gravitating scalar field are not stable in spherical symmetry and therefore the cosmic censorship conjecture is true in this context. The original proof is by contradiction and sharp estimates are obtained strictly depending on spherical symmetry. In this paper, appropriate a priori estimates for the solution are obtained. These estimates are more relaxed but sufficient for giving another robust argument in proving the instability, in particular not by contradiction. In another related paper, we are able to prove instability theorems of the spherical symmetric naked singularities under certain isotropic gravitational perturbations without symmetries. The argument given in this paper plays a central role.